Question 1 a) The simulation models of verification and
validation is of great significance. What do you think would be the
drive for model verification? Compare and contrast the issues
involved in verification and validation AN (8marks)
b) Is very necessary to carryout an output analysis of
simulation model. Why? Are the output data from simulation normally
distributed? Discuss your answer with example AN (7marks)
c) Suppose you are developing the simulation model of a
communication system where the incoming calls are being served by a
single service person. Assume the arrival of the calls as well as
their service times are random. Identify some entities, attributes,
activities, events and states variables for this simulation model.
AP (5 marks) TOTAL:20Marks
Question 2 a) Describe the linear congruential method for Random
number generation. Using the multiplicative congruential method,
generate a sequence of fivethree-digit random integers with seed =
117 constant multiplier = 43 and modulus = 1000 AP (5 marks)
3 b) With your own understanding, describe the Mid-Square method
of generating Pseudo–Random Numbers and generate 5 Pseudo–Random
Numbers from the seed number 4172 CR (5 marks) c) Suppose that X
and Y are jointly continuous random variable EV (10 marks)
TOTAL:20Marks
Question 3 a) Why is the Markov property useful in modelling
queueing systems? AN (5 marks)
b) A telephone exchange multiplexes 64 Kb/s voice calls onto a
256 Kb/s trunk line (therefore the line will hold at most four
calls). New calls have an exponentially distributed inter-arrival
process, with a mean of 20 seconds, and the call holding time is
exponentially distributed with a mean of 60 seconds.
(i) Draw a diagram of a Markov Chain which models the system,
labelling the state transitions with their rates where
appropriate.
(ii) What is the necessary condition for stability of this
system? AP (5 marks)
(c.) Assume that the probability of rain tomorrow is 0.5 if it
is raining today, and assume that the probability of its being
clear (no rain) tomorrow is 0.9 if it is clear today. Also assume
that these probabilities do not change if information is also
provided about the weather before today.
(i) Explain why the stated assumptions imply that the Markovian
property holds for the evolution of the weather.
4 (ii) Formulate the evolution of the weather as a Markov chain
by defining its states and giving its (one-step) transition matrix
CR (10 marks) TOTAL:20Marks
Question 4 Consider a birth death model with birth rates in
states n = 0, 1, . . . and death rates in states n = 1, 2, . .
(a) Give the detailed balance conditions for the equilibrium
probability of being in state n, for n = 0, 1, . . . and explain
how they are derived. AN (5 marks)
(b) Using the detailed balance conditions derive an expression
and state any condition needed to ensure that the equilibrium
distribution exists. EV (5marks)
(c) i Now consider the M/M/1 queue with arrival rate ? and
service rate µ and explain how it can be modelled as a birth death
model.
ii. For the M/M/1 queue derive the form of , the equilibrium
distribution of the number of customers present, and give any
conditions needed to ensure the existence of the equilibrium
distribution. Derive the mean value of the equilibrium distribution
EV(10 marks) TOTAL:20Marks
Question 5 a) Distinguish between continuous and discrete system
AN (5 marks)
Papaye fast food has a one check out point. Customers arrive at
this checkout at random from 1 to 8 minutes apart and each interval
time as the same probability 5 of occurrence. The service time vary
from 1 to 6 minutes, with probability given below Service (mins) 1
2 3 4 5 6 Probability 0.10 0.20 0.30 0.25 0.10 0.05 b) Simulate the
arrival of 6 customers CR (10 marks)
c) From your simulation, compute the following (i) Average
waiting time for a customer
(ii) The probability that a customer has to wait (iii) The
probability of a customer being idle
(iv) The average service time (v) The average time between
arrival EV (5 marks)
Use the following sequence of random numbers:
Random digit for arrival: 913, 727, 015, 948, 309, 922
Random digit for service time: 84, 10, 74, 53, 17, 79
TOTAL:20Mark